Given a matrix A over field F of order 5x5, with tr(A)=0, and p(A)=1, find the eigenvalues of A.
If the question would have been over C/R I would have found an answer, but this F field thing is too much for me...
Observe that . If you still cant see it, observe that is a scalar.
Answer to question 1:
We will show that any eigenvector corresponding to a non-zero eigenvalue is a scalar multiple of x.
Let us say it had an eigenvector z, corresponding to a eigen value ,then , Thus if , z is in the span of x.